| L(s) = 1 | + 128·2-s − 618.·3-s + 1.63e4·4-s + 1.79e4·5-s − 7.91e4·6-s + 2.09e6·8-s − 1.39e7·9-s + 2.30e6·10-s − 3.48e7·11-s − 1.01e7·12-s − 1.86e8·13-s − 1.11e7·15-s + 2.68e8·16-s − 9.77e6·17-s − 1.78e9·18-s − 3.06e9·19-s + 2.94e8·20-s − 4.46e9·22-s + 2.91e10·23-s − 1.29e9·24-s − 3.01e10·25-s − 2.39e10·26-s + 1.74e10·27-s + 5.56e9·29-s − 1.42e9·30-s + 2.63e11·31-s + 3.43e10·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.163·3-s + 0.5·4-s + 0.102·5-s − 0.115·6-s + 0.353·8-s − 0.973·9-s + 0.0727·10-s − 0.539·11-s − 0.0815·12-s − 0.825·13-s − 0.0167·15-s + 0.250·16-s − 0.00577·17-s − 0.688·18-s − 0.785·19-s + 0.0514·20-s − 0.381·22-s + 1.78·23-s − 0.0576·24-s − 0.989·25-s − 0.583·26-s + 0.321·27-s + 0.0599·29-s − 0.0118·30-s + 1.71·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(2.598456005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.598456005\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 128T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 618.T + 1.43e7T^{2} \) |
| 5 | \( 1 - 1.79e4T + 3.05e10T^{2} \) |
| 11 | \( 1 + 3.48e7T + 4.17e15T^{2} \) |
| 13 | \( 1 + 1.86e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 9.77e6T + 2.86e18T^{2} \) |
| 19 | \( 1 + 3.06e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 2.91e10T + 2.66e20T^{2} \) |
| 29 | \( 1 - 5.56e9T + 8.62e21T^{2} \) |
| 31 | \( 1 - 2.63e11T + 2.34e22T^{2} \) |
| 37 | \( 1 + 1.93e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 2.63e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 6.20e11T + 3.17e24T^{2} \) |
| 47 | \( 1 - 4.22e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 7.45e12T + 7.31e25T^{2} \) |
| 59 | \( 1 - 2.77e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 2.13e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 4.66e12T + 2.46e27T^{2} \) |
| 71 | \( 1 - 3.38e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 6.72e12T + 8.90e27T^{2} \) |
| 79 | \( 1 + 2.25e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 2.65e14T + 6.11e28T^{2} \) |
| 89 | \( 1 - 8.87e13T + 1.74e29T^{2} \) |
| 97 | \( 1 - 9.04e14T + 6.33e29T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.18847411444724744356285363221, −10.20611124363073088006113021862, −8.847133729306202168768134572516, −7.66432324711468260367460578303, −6.46880859709876771198947893882, −5.44022787642588935705916720098, −4.53928623165916601637911127705, −3.06956821362095662087288716981, −2.25743869991537477433508320380, −0.63156976591671227814248399862,
0.63156976591671227814248399862, 2.25743869991537477433508320380, 3.06956821362095662087288716981, 4.53928623165916601637911127705, 5.44022787642588935705916720098, 6.46880859709876771198947893882, 7.66432324711468260367460578303, 8.847133729306202168768134572516, 10.20611124363073088006113021862, 11.18847411444724744356285363221
